Learning ObjectivesIn this chapter you will learn

the basic rules of probabilityabout estimating the probability of the occurrence of an event

the Central Limit Theoremhow to establish confidence intervals

Types of Probability

Three approaches to probabilityMathematicalEmpiricalSubjective

Mathematical Probability

Mathematical (or classical) probabilitybased on equally likely outcomes that can be calculated

useful when equal chance of outcomes and random selection is possible

Example20 people are arrested for crimes 2 are innocentIf one of the accused is picked

randomly, what is the probability of selecting and innocent person?

Solution2/20 or .1 – 10% chance of picking an

innocent person

Empirical ProbabilityEmpirical probability

uses the frequency of past events to predict the future

calculated the number of times an event occurred

in the past divided by the number of observations

Example75,000 autos were registered in the county last year650 were reported stolenWhat is the probability of having a car stolen this year?

Solution650/75,000 .009 or .9%

Subjective ProbabilitySubjective probabilitybased on personal reflections of an individual’s opinion about an event

used when no other information is available

ExampleWhat is the probability that Al Gore will win the next presidential election? Obviously, the answer depends on who you ask!

Probability RulesWe sometimes need to combine the probability of events two fundamental methods of combining probabilities areby additionby multiplication

The Addition RuleThe Addition Rule

if two events are mutually exclusive (cannot happen at the same time)

the probability of their occurrence is equal to the sum of their separate probabilities

P(A or B) = P(A) + P(B)

ExampleWhat is the probability that an odd number will result from the roll of a single die?6 possible outcomes, 3 of which are odd numbers

Formula 50.2

1

6

1

6

1

6

1==++

The Multiplication Rule

Suppose that we want to find the probability of two (or more events) occurring

together?

The Multiplication Ruleprobability of events are NOT mutually exclusive equals the product of their separate probabilitiesP(B|A) = P(A) times P(B|A)

ExampleTwo cards are selected, without replacement, from a standard deckWhat is probability of selecting a 10 and a 4?

P(B|A) = P(A) times P(B|A)006.

2652

16

51

4

52

4≈=•

Laws of ProbabilityThe probability that an event will

occuris equal to the ratio of “successes”

to the number of possible outcomes the probability that you would flip a

coin that comes up “heads” is one out of

two or 50%

Gambler’s FallacyProbability of flipping a head extends to the next toss and every toss

thereafter mistaken belief that

if you tossed ten heads in a rowthe probability of tossing another is

astronomicalin fact, it has never changed – it is still 50%

Calculating Probability

You can calculate the probability of any given total that can be thrown in a game of “Craps”each die has 6 sideswhen a pair of dice is thrown, there are how many possibilities?

Die #1 Roll

Die #2Roll

1 2 3 4 5 6

1 2 3 4 5 6 72 3 4 5 6 7 83 4 5 6 7 8 94 5 6 7 8 9 105 6 7 8 9 10 116 7 8 9 10 11 12

Outcomes of Rolling Dice

Number of Ways to Roll Each Total

Total Roll N of Ways 2 1 3 2 4 3 5 4 6 5 7 6 8 5 9 4

10 3 11 2 12 1

Winning or Not?What is the probability of….losing on the first roll?

1/36 + 2/36+ 1+36 = 4/36 or 11.1%

rolling a ten? 3/36 or 1/12 = 8.3%

Next Rollmaking the point on the next

roll?now we calculate probability

P(10) + P(any number, any roll) = 1/3(1/12) times (1/3) = 2.8%

Making the Point The probability of making the

point for any number to calculate this probabilityuse both the Addition Rule and

the Multiplication Rulethe probability of two events that are not

mutually exclusive are the product of their separate probability

ContinuingAdd the separate probabilities of rolling each type of numberP(10) x P (any number, any roll) = 1/12

x 1/3 = 1/36 or 2.8% is the P of two 10s or two 4s

P of two 5s or two 9s = (1/9) (2/5) = 2/45 = 4.4%

P of two 6s or two 8s = (5/36) (5/11) = 25/396 = 6.3%

Who Really Wins?Add up all the probabilities of

winning(2/9) + 2 (1/36) + 2 (2/45) + 2

(25/396) = (2/9) + (4/45) + (25/198) = 244/495 or 49.3%

What is the probability that you will lose in the long run or that the Casino wins?

Empirical ProbabilityEmpirical probability is based

upon research findingsExample: Study of Victimization Rates among American Indians

Which group had the greatest rate of violent crime victimizations?

The lowest rate?

Violent Crime Victimization By Age, Race, & Sex of Victim, 1992

- 1996

Percent of Violent Crime Victimization

VictimAge/Sex

AmericanIndian White Black Asian Total

12 – 17 20.4% 23.8% 26.8% 24.0% 24.2%18 – 24 31.5 23.4 24.0 21.7 23.625 – 34 23.5 23.6 23.2 26.3 23.635 – 44 18.0 17.1 16.6 18.3 17.045 – 54 4.7 7.8 6.1 7.3 7.5

55 & Older 1.9 4.3 3.3 2.4 4.1MALE 58.9 58.4 50.5 62.6 57.4

FEMALE 41.1 41.6 49.5 37.4 42.6

Highest rate by race & age

Lowest rate by race & age

Using ProbabilityWe use probability every day

statements such aswill it may rain today?will the Red Sox win the World Series?will someone break into my house?

We use a model to illustrate probabilitythe normal distribution

The Normal Distribution

μ +2σ-2σ +1σ +3σ-1σ-3σ

Approximately 68% of area under the curve falls with 1 standard deviation from the mean68.26%

| 95.44% |

| 99.72% |

Approximately 1.5% of area

falls beyond 3 standard deviations

Z ScoresThe standard score, or z-scorerepresents the number of standard

deviations a random variable x falls from the

mean μ

σμ−

==x

zdeviation standard

mean - value

ExampleThe mean of test scores is 95

and the standard deviation is 15find the z-score for a person who scored an 88

Solution 467.015

9588−≈

−

Example ContinuedWe then convert the z-score into

the area under the curvelook at Appendix A.2 in the textthe fist column is the first & second values of z (0.4)

the top row is the third value (6)cumulative area = .3228

Another Use of Probability

We can also take advantage of probability when we draw samplessocial scientists like the properties of the normal distribution

the Central Limit Theorem is another example of probability

The Central Limit Theorem

If repeated random samples of a given size are drawn from any population (with a mean of μ

and a variance of σ)then as the sample size becomes largethe sampling distribution of sample

means approaches normality

ExampleRoll a pair of

dice 100 timesThe shape of

the distribution of outcomes will

resemble this figure

Dot/Lines show counts

2.5 5.0 7.5 10.0

v1

0

5

10

15

Standard Error of the Sample Means

The standard error of the sample means is the standard deviation of the sampling distribution of the sample means

σσ

x n=

Standard Error of the Sample Means

If σ is not known and n 30 the standard deviation of the sample, designated s

is used to approximate the population standard deviation

the formula for the standard error then becomes:

ss

nx =

Confidence IntervalsAn Interval Estimate states the

range within which a population parameter probably liesthe interval within which a

population parameter is expected to occur is called a confidence interval

two confidence intervals commonly used are the 95% and the 99%

Constructing Confidence Intervals

In general, a confidence interval for the mean is computed by:

X Zs

n±

95% and 99% Confidence Intervals 95% CI for the population mean is calculated by

Xs

n±196.

Xs

n±258.

99% CI for the population mean is calculated by

SummarySocial scientists use probability

to calculate the likelihood that an event will occur

in various combinationsfor various purposes (estimating a population parameter, distribution of scores, etc.)